Partially inspired by the idea of ESAV method [1]},
we rewrite the equation (REF ) as the following equivalent system:
\(u_t & = \varepsilon ^2\Delta u + \frac{\exp \lbrace s\rbrace }{\exp \lbrace E_1(u)\rbrace } f(u), \\s_t & = - \frac{\exp \lbrace s\rbrace }{\exp \lbrace E_1(u)\rbrace } (f(u),u_t).\) 
